Example of Quasi-circular domain

Question

Can someone provide me with an example of a quasi-circular domain?

A domain $D\subset\mathbb{C}^n$ is said to be m-quasi-circular, (where $m=(m_1,m_2,..,m_n); m_i$ being positive integers), if D is invariant under the map $f_{m,\theta}$ for all $\theta \in \mathbb{R}$. $$f_{m,\theta} : D\rightarrow \mathbb{C}^n$$ $$(z_1,\dots,z_n)\mapsto (e^{im_1\theta}z_1,\dots,e^{im_n\theta}z_n)$$

Note: If $m_1=m_2=m_3=....=m_n=1$, then the domain is called circular domain.

Can you give an example of a quasi-circular domain that is not circular? Say (2,3)-quasi-circular domain in $\mathbb{C}^2$.

Answer 1

Symmetrized bidisc is a quasi-circular domain that isn't a circular domain.

$$\mathbb{G}=\{(z_1+z_2,z_1z_2)\in \mathbb{C}^2: z_1,z_2\in B(0,1)\}$$ Clearly, this is a (1,2)-quasi-circular domain.

To see that this is not a circular domain, notice that $(2,1)\in \bar{\mathbb{G}}$ but $(-2,-1)\not \in \overline{\mathbb{G}}$.